Where 360 Came From — and Why the Circle Never Agreed
In the previous guide the unit circle set trigonometry free: every angle, however large or negative, became a point on a circle of radius one, and sin and cos became that point's height and shadow. But all through that guide we kept measuring angles in degrees, where one full turn is 360 of them. Stop and ask the obvious question nobody asks: why 360? Not because of anything in the circle. The number is an inheritance from Babylonian astronomers, who liked 360 because it is close to the days in a year and divides neatly into halves, thirds, quarters, fifths, and sixths. It is a calendar's number, bolted onto geometry.
Look at what that arbitrariness costs. From the mensuration rung you know a circle's circumference is 2 pi r — the radius r appears, and so does pi, but 360 is nowhere in sight. The circle's own geometry is written in radii and pi. So whenever we describe a turn with degrees, we are speaking a foreign language and paying a translation tax: a fudge factor of pi/180 has to be smuggled into formula after formula to convert between the circle's natural units and our calendar's. The whole point of this guide is to drop the foreign currency and let the circle pay in its own coin.
The Definition: An Angle Measured by Its Own Arc
Here is the idea in one picture. Stand at the centre of a circle of radius r and open up an angle. That angle cuts off an arc along the rim. Now ask: how long is that arc, measured in radii? If the arc happens to be exactly one radius long, we call the angle one radian. If the arc is two radii long, the angle is two radians; if half a radius, half a radian. The radian measure of an angle is simply the arc it subtends divided by the radius — a pure ratio of two lengths, arc over radius, with the units of length cancelling out.
Because it is a ratio of two lengths, a radian is dimensionless — it carries no metres or inches, it is just a number. This is the quiet beauty of the definition: it does not depend on how big the circle is. A small circle and a huge circle, opened to the same angle, cut off arcs of very different lengths, but each arc is the same number of radii, because both arc and radius scale together. The angle is captured by the shape of the opening alone, exactly as an angle should be.
radian measure of an angle = arc length / radius = s / r so, turned around: arc length s = r * theta (theta in radians)
The Number That Replaces 360
Now do the most important calculation in the guide: how many radians is one full turn? Going all the way around, the arc is the entire circumference, 2 pi r. Divide by the radius and the r cancels, leaving 2 pi. So a full turn is 2 pi radians — about 6.28 — and that single fact is the whole conversion table. A half turn (a straight angle, the old 180 degrees) is pi radians; a quarter turn (a right angle, 90 degrees) is pi/2; an eighth of a turn (45 degrees) is pi/4. Notice these are exact: no decimals, no rounding, just pi and small fractions.
From here, converting either way is one ratio. Since 180 degrees equals pi radians, multiply degrees by pi/180 to get radians, or multiply radians by 180/pi to go back. That pi/180 is exactly the 'translation tax' from the first section — and now you can see it is nothing mysterious, just the exchange rate between two units for a turn. The reason mathematicians keep angles in radians is that this exchange rate then never reappears: in radians, the formulas of the circle and of calculus carry no stray pi/180 at all.
- Convert 60 degrees to radians: multiply by pi/180. So 60 times pi/180 = pi/3 radians. Reading it as a fraction of a turn, pi/3 is one sixth of 2 pi, and indeed 60 degrees is one sixth of 360 — the two agree.
- Convert 270 degrees: 270 times pi/180 = 3 pi/2 radians, which is three quarters of a full turn — exactly the angle pointing straight down on the unit circle.
- Convert 2 radians back to degrees: 2 times 180/pi, which is about 114.6 degrees. Not a round number of degrees — a clean radian angle is usually a messy degree angle, and that is exactly the point.
The Payoff: Arc, Sector, and a Formula Made Trivial
Watch the units earn their keep. Rearranging the definition gives arc length s = r times theta — to find how far you have travelled along the rim, just multiply the radius by the angle in radians, full stop. Compare the degree version, where you must write (n/360) times 2 pi r and shuffle the 360 around. The radian formula has no 360, no separate 2 pi to drag along; the geometry of the circle speaks in one clean line. The same simplicity reaches the area of a sector: in radians it is one-half times r^2 times theta, where the degree version needed (n/360) times pi r^2.
These two formulas are not new geometry — they are exactly the mensuration-rung results, just with the angle measured by its own arc instead of by 360ths. But the deepest payoff is one you cannot fully prove until the calculus rungs: when the angle is in radians, sin theta is almost exactly equal to theta for small angles. Try it — sin of 0.01 radians is 0.0099998..., breathtakingly close to 0.01 itself. This near-equality, written sin theta is about theta, is the seed from which the derivative of sin grows into cos, and it is true only in radians. In degrees, sin of 0.01 degrees is about 0.0001745, and the clean relationship is buried under a factor of pi/180.
Reading Familiar Angles in the New Units
The unit circle you built last guide does not change at all — only the labels around its rim do. The points stay put; we relabel each one by how much arc you walked to reach it, starting from the positive x-axis and going counterclockwise. The familiar special angles 30, 45, 60 degrees become pi/6, pi/4, pi/3, and their multiples sweep around: pi/2 at the top, pi at the far left, 3 pi/2 at the bottom, 2 pi back home. A reference angle works identically — you still fold any angle back to its acute partner against the x-axis, only now that partner is named in radians, like pi/6 instead of 30 degrees.
Radians also make the graphs honest. When you plot the graph of sin with the angle in radians along the horizontal axis, one full wave completes after exactly 2 pi, and the curve crosses zero at 0, pi, 2 pi — clean landmarks. Better still, near the origin the sine graph leaves at a slope of exactly 1, a direct consequence of sin theta being about theta; in degrees that starting slope is the ugly pi/180 instead. The wave is the same shape either way, but only radians let it start at the natural 45-degree tilt.
What You Carry Forward
You have not learned a new kind of angle — you have learned a better ruler for the angles you already had. A radian is an angle measured by the arc it cuts, in units of the radius; a full turn is 2 pi of them, and the conversion to degrees is the single fact 180 degrees = pi radians. The reward is that arc length, sector area, the graphs of trig functions, and the small-angle behaviour all come out clean, with no calendar number 360 fouling the algebra. From here on, unless a problem hands you degrees, assume an angle is in radians.
Next, guide 4 leaves the right triangle and the unit circle behind to tackle any triangle at all, with the law of sines and the law of cosines. Those laws are usually written and proved in radians, and the triangle-area-from-sine formula you will meet there is one more place the radian's cleanliness quietly pays off. Keep the conversion 180 degrees = pi radians in your pocket, and keep the picture of an arc one radius long — that picture is the whole idea.